Landlord's Malfunctioning AC system created high utility bills: how to make them help cover the cost?

Location: Illinois (Coles County)

I rent an apartment with a central air system, and have had increasingly high electric bills over the past few months driven by crazy high usage. We had been asking them to fix the AC for months, and they had managed to make it work a little better but had not fixed the actual usage problem.

It came to a head when we used 5600 kwh in July when the same time last year we used 1000 kwh (same weather). After presenting that data and some back and forth in which the property manager and landlord were dismissive, the property manager informed us that they would not be investigating this and that our lease would not be renewed (expires in October). I called the landlord directly at this point and they walked that back, saying "I just thought since you're so unhappy here, you would want to move" (sure). I convinced the landlord to send an HVAC technician, who discovered a material problem (intermittent sequencers defect).

With that fixed, the AC is working much more efficiently but we are still on the hook for this ridiculous electric bill for July. Please advise how best to persuade the landlord to help cover the cost of this bill. What would be a fair request? If possible, we would love to avoid any actual legal proceedings but are prepared to go there as a last resort.